Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

§1. INTRODUCTIONSeveral physically meaningful systems of ordinary differential equationsare of second order, as for instance the free particle in m-dimensional space,the damped or undamped harmonic oscillator, coupled or not, having constanttime-dependent frequency or not, etc. Such systems are ubiquitous inNewtonian Mechanics, in Hamiltonian Dynamics and in General Relativity.Two classical major problems are to classify these systems up to point orcontact equivalence (Lie’s Grail) and to recognize when they coincide withthe Euler equations associated to a Lagrangian (inverse variational problem).In small dimensions, complete results hold (Lie, Tresse, Cartan; Darboux,Douglas). However, in arbitrary dimension, both tasks quickly exceed thehuman as well as the digital computer scale, due to the intrinsic complexityof the underlying symbolic computations (explosion, swelling) and to theexponentially increasing number of cases to be treated. We refer to Olver’smonograph [Ol1995] for a panorama of problems, methods and results.At least, as a first step in classification, with respect to applications, thereare both a mathematical and a physical interest in determining concrete, explicitand applicable (“ready-made”) criteria for a system of ordinary differentialequation to be equivalent, via a local point transformation, to a linearequation.31.1. Scalar equation. In this respect, we remind the celebrated linearizabilitycriterion for a single equation, due to Lie. Let K = R of C. Let x ∈ Kand y ∈ K. Consider a local second order ordinary differential equationy xx = F(x, y, y x ), possibly nonlinear, with a locally K-analytic right-handside 1 .Theorem 1.2. ([Lie1883], pp. 362–365; [GTW1989]; [Ol1995], p. 406) Thefollowing four conditions are equivalent:(1) y xx = F(x, y, y x ) is equivalent under a local point transformation(x, y) ↦→ (X, Y ) to the free particle equation Y XX = 0;(2) y xx = F(x, y, y x ) is equivalent to some linear equation Y XX =G 0 (X) + G 1 (X) Y + H(X) Y X ;(3) the local Lie symmetry group of y xx = F(x, y, y x ) is eightdimensional,locally isomorphic to the group PGL(3, K) of allprojective transformations of P 2 (K).1 By borrowing techniques developed in [Ma2003], this theorem as well as the next bothhold under weaker smoothness assumptions, namely with a C 2 or a W 1,∞locright-hand side.